Welcome to the World of Multiple Lives!
In your CM1 journey so far, you have mostly looked at models involving a single person (a "single life"). But in the real world, many insurance products involve couples, business partners, or families. This chapter introduces multiple life models, specifically focusing on how we calculate premiums and values when two lives are involved.
Don't worry if this seems a bit overwhelming at first! If you understand single-life functions, you already have the building blocks. We are just going to learn how to combine them. Think of it like moving from a solo performance to a duet—the music is richer, but the basic notes are the same.
1. The Two Basic Statuses: Joint Life and Last Survivor
When dealing with two lives, say (x) and (y), we usually care about one of two scenarios: when the first person dies or when the last person dies.
The Joint Life Status \( (xy) \)
The joint life status exists only as long as both individuals are alive. It "fails" or ends the moment the first person dies.
Analogy: Think of a doubles tennis match. The "team" status ends as soon as one player leaves the court.
- Notation: We write this simply as \( xy \).
- Probability of Survival: For the status to survive \( t \) years, both \( x \) and \( y \) must survive. Assuming their lives are independent: \( {}_tp_{xy} = {}_tp_x \times {}_tp_y \).
The Last Survivor Status \( (\overline{xy}) \)
The last survivor status exists as long as at least one of the two individuals is alive. It only "fails" when the second (last) person dies.
Analogy: Think of a "Buy One, Get One Free" voucher that stays valid as long as you have at least one item left in your bag.
- Notation: We write this with a bar over the ages: \( \overline{xy} \).
- Probability of Survival: This is the probability that either \( x \) is alive, OR \( y \) is alive, OR both are alive. It is easier to calculate as \( 1 - (\text{Probability both are dead}) \).
- Formula: \( {}_tp_{\overline{xy}} = {}_tp_x + {}_tp_y - {}_tp_{xy} \).
Quick Review:
Joint Life \( (xy) \) = Ends at First death.
Last Survivor \( (\overline{xy}) \) = Ends at Second death.
2. Assurance Functions for Two Lives
Just like single lives, we can have life assurance contracts that pay out based on these statuses.
Joint Life Assurance \( A_{xy} \)
This pays out a benefit of 1 at the end of the year of the first death (or immediately if it is \( \bar{A}_{xy} \)). It doesn't matter who dies first; the money is paid as soon as the first person passes away.
Last Survivor Assurance \( A_{\overline{xy}} \)
This pays out a benefit of 1 at the end of the year of the second death. This is common for "whole of life" policies used for inheritance tax planning.
The Key Relationship Formula
There is a very important identity you must memorize. It links single lives, joint lives, and last survivors:
\( A_x + A_y = A_{xy} + A_{\overline{xy}} \)
Why? Think of it this way: if you have two policies—one that pays on the first death and one that pays on the second death—you are essentially guaranteed two payouts in total, one for \( x \)'s death and one for \( y \)'s death.
Key Takeaway: If you know any three of these values, you can always find the fourth!
3. Annuity Functions for Two Lives
Annuities follow the same logic as assurances. We just need to decide when the payments stop.
Joint Life Annuity \( a_{xy} \)
Payments are made as long as both are alive. Payments stop the moment either person dies.
Last Survivor Annuity \( a_{\overline{xy}} \)
Payments continue as long as at least one person is alive. This is the most common type of pension for couples; if one spouse dies, the survivor keeps receiving the income.
The Formula:
Just like with assurances, we have a beautiful relationship:
\( a_{\overline{xy}} = a_x + a_y - a_{xy} \)
Step-by-Step Calculation Tip:
1. Calculate the single life annuities \( a_x \) and \( a_y \).
2. Calculate the joint life annuity \( a_{xy} \).
3. Use the formula above to find the last survivor annuity.
4. Contingent Assurances
Sometimes, a benefit is only paid if a specific person dies first. We use a small number "1" over the age of the person who must die first to trigger the payment.
- \( A_{xy}^1 \): A benefit paid on the death of \( x \), provided \( x \) dies before \( y \).
- \( A_{xy}^2 \): A benefit paid on the death of \( x \), provided \( x \) dies after \( y \).
Did you know?
If you add \( A_{xy}^1 \) and \( A_{yx}^1 \), you get \( A_{xy} \). This is because the first person to die must be either \( x \) or \( y \)!
Common Mistake: Don't confuse \( A_{xy}^2 \) with \( A_{\overline{xy}} \). \( A_{\overline{xy}} \) pays when the second person dies (whoever that is). \( A_{xy}^2 \) pays when \( x \) dies, but only if they are the second one to go.
5. Reversionary Annuities
A reversionary annuity is a special type of "waiting" annuity. It is often written as \( a_{x|y} \). This means an annuity is paid to \( y \), but only after \( x \) has died.
Analogy: Imagine a relay race. \( y \) is waiting for \( x \) to "pass the baton" (die) before \( y \) can start running (receiving payments). If \( y \) dies before \( x \), the annuity never starts!
The Simple Logic:
The value of a reversionary annuity to \( y \) after \( x \) is simply the value of a full annuity for \( y \), minus the time they were both alive.
\( a_{x|y} = a_y - a_{xy} \)
6. Evaluation under the Independence Assumption
To solve these problems in exams, we usually assume the two lives are independent. This makes the math much easier.
Key Rules for Independence:
- Survival Probabilities: \( {}_tp_{xy} = {}_tp_x \times {}_tp_y \).
- Force of Mortality: The force of mortality for a joint life status is the sum of the individual forces: \( \mu_{xy}(t) = \mu_x(t) + \mu_y(t) \).
- Expected Present Value (EPV): If both lives follow a Constant Force of Mortality (say \( \mu_x \) and \( \mu_y \)) and the interest rate is \( \delta \), then:
\( \bar{a}_{xy} = \frac{1}{\mu_x + \mu_y + \delta} \)
Memory Trick: For joint lives, the "risk" of the status ending is higher because there are two people who could die. This is why we add the forces of mortality together!
7. Summary & Quick Review
Key Formulas to Memorize:
- Last Survivor Annuity: \( a_{\overline{xy}} = a_x + a_y - a_{xy} \)
- Reversionary Annuity: \( a_{x|y} = a_y - a_{xy} \)
- Probability (Joint): \( {}_tp_{xy} = {}_tp_x \times {}_tp_y \)
- Probability (Last Survivor): \( {}_tp_{\overline{xy}} = {}_tp_x + {}_tp_y - ({}_tp_x \times {}_tp_y) \)
Final Encouragement: Multiple life functions are just about understanding who needs to be alive for a payment to happen. Practice drawing timelines if you get stuck—mark when \( x \) dies and when \( y \) dies, and shade the areas where payments occur. You've got this!